With respect to classical Lebesgue or Hardy spaces, in the case \(m>1,\)  in
the paper [1]}, we obtained the \(H^{p}(\mathbb {R}^{n})-L^{q}(\mathbb {R}^{n})\)  boundedness of these operators and we show that we cannot expect the \(H^{p}(\mathbb {R}^{n})-H^{q}(\mathbb {R}^{n})\)  boundedness of them.This is an
important difference with the case \(m=1.\)  Indeed, in the paper [2]},
M. Taibleson and G. Weiss, using the molecular characterization of the real
Hardy spaces, obtained the boundedness of \(I_{\alpha }\)  from \(H^{p}(\mathbb {R}^{n})\)  into \(H^{q}(\mathbb {R}^{n}),\)  \(0<p\le 1.\)  In this paper we will
extend both results to the setting of variable exponents. Here and below we
shall postulate the following conditions on \(p(.),\)
\(\left|p(x)-p(y)\right|\le \frac{c}{-\log \left|x-y\right|},\text{ }\left|x-y\right|<\frac{1}{2},\) 
